The present invention relates to a computerized method for simulating a rubber compound comprising a rubber matrix and filler particles which serves to estimate the loss tangent values for a short time.
In general, in order to know properties such as loss tangent values of a rubber compound, the rubber compound is measured for its strain and stress by the use of a viscoelasticity tester, and a loss tangent value is determined from the relationship between the strain and stress. In the measurement, a strain (a) is applied to the rubber compound, wherein the strain (a) has a fixed semi-amplitude and a fixed frequency and the strain (a) varies as shown in FIG. 12, and the stress (b) caused by the strain is measured. To give a more concrete example, a test specimen of a rubber compound is applied by an initial tensile stress of 10%, and the test specimen is further applied by strain which varies sinusoidally with a semi-amplitude of 1% and a frequency of 10 Hz. under such condition, the resultant stress is measured, wherein the stress measured is varied sinusoidally similarly to the strain but there is a phase lag (phase difference δ).
using this phase lag or difference δ and also the semi-amplitude σ0 of the stress (b) and the semi-amplitude ε0 of the strain (a), the loss tangent can be determined as follows.Storage elastic modulus E′=(σ0/ε0)·cos δLoss elastic modulus E″=(σ0/ε0)·sin δLoss tangent δ=E″/E′
In recent years, on the other hand, in order to design and develop a rubber compound comprising a rubber matrix and filler particles, computerized methods for simulating a rubber compound have been proposed for example as disclosed in Japanese Patent Application Publication Nos. 2006-175937 and 2009-259043.
In such a computer simulation, as the rubber molecular chain and filler can be included in the simulation calculation, it becomes possible to estimate properties of the rubber compound without producing a number of rubber compounds experimentally. Therefore, by utilizing the above-mentioned simulation, it becomes possible to predict how the properties of a rubber compound are changed by employing different compositions with respect to filler contents such as carbon and silica for example, without actually producing various rubber compounds.
In the above-mentioned simulation methods, it is impossible to provide a continuously-varying sinusoidal strain (a) as a deformation condition for the rubber compound model. Therefore, as shown in FIG. 11(a), the strain whose amplitude varies sinusoidally is provided discretely by a fixed time increment Δt, and for each discrete strain, the stress is computed through a convergence calculation technique. As a result, the stress response as shown in FIG. 11(b) as an example is obtained.
On the other hand, in order to obtain the loss tangent value, it is necessary to know the above-mentioned phase lag or difference. In order to obtain the phase difference accurately, a continuously-changing stress curve is necessary in determining the time when the amplitude σ0 occurs. Heretofore, therefore, a sine wave most fitting to the computed discrete stress values is obtained as an approximate waveform of the stress, and the time when the amplitude ν0 occurs is determined by the use of the approximate waveform.
In the above-mentioned simulation methods, in order to improve the accuracy of the approximate waveform of the stress, the computation of the stress value has to be made frequently as shown in FIG. 11(b). In other words, the number of the computations per one cycle has to be increased. Further, to obtain a stable computational value of the phase difference between the stress and strain, the computations have to be made over multiple cycles. Thus, the computational time and cost increase.